Question: The graph of a sinusoidal function intersects its midline at $(0,5)$ and then has a maximum point at $(0.75,7)$. Write the formula of the function, where $x$ is entered in radians. $f(x)=$
Solution: The strategy First, let's use the given information to determine the function's amplitude, midline, and period. Then, we should determine whether to use a sine or a cosine function, based on the point where $x=0$. Finally, we should determine the parameters of the function's formula by considering all the above. Determining the amplitude, midline, and period The midline intersection is at $y={5}$, so this is the midline. The maximum point is $2$ units above the midline, so the amplitude is ${2}$. The maximum point is $0.75$ units to the right of the midline intersection, so the period is $4\cdot 0.75={3}$. [Why did we multiply by 4?] Determining the type of function to use Since the graph intersects its midline at $x=0$, we should use the sine function and not the cosine function. This means there's no horizontal shift, so the function is of the form $a\sin(bx)+d$. [How do we know that?] Determining the parameters in $a\sin(bx)+d$ Since the midline intersection at $x=0$ is followed by a maximum point, we know that $a>0$. [How do we know that?] The amplitude is ${2}$, so $|a|={2}$. Since $a>0$, we can conclude that $a=2$. The midline is $y={5}$, so $d=5$. The period is ${3}$, so $b=\dfrac{2\pi}{{3}}$. The answer $f(x)=2\sin\left(\dfrac{2\pi}{3}x\right)+5$